On some groups of automorphisms of von Neumann algebras with cyclic and separating vector

Abstract

Let\(\mathfrak{A}\) be a von Neumann algebra with the vector ω cyclic and separating for\(\mathfrak{A}\). Let\(\mathfrak{B}_G\) be a group of unitary operators under which both ω and\(\mathfrak{A}\) are invariant. Let\(\mathfrak{B}\) (resp. ℜ′) be the fixed point algebra in 21 (resp. in\(\mathfrak{A}\)′). LetFo be an orthogonal projection onto the subspace of all vectors invariant under\(\mathfrak{B}_G\). It is shown that ℜ=(\(\mathfrak{A}\) ν {Fo})″ and that the irreducibility of ℜ implies thatFo is one-dimentional. Other consequences of the Theorem ofKovacs andSzucs are also derived. In sec. 3. the spectrum properties of the group\(\mathfrak{B}_G\) are studied. It is proved that the point spectrum of\(\mathfrak{B}_G\) is symmetric and that it is a group provided ℜ is irreducible. In this case there exists a homomorphism χ→\(\hat \chi\) (resp. χ →\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\chi }\)) of the point spectrum of\(\mathfrak{B}_G\) into the group of unitary operators in\(\mathfrak{A}\) (resp. in\(\mathfrak{A}\)′) uniquely (up to the phase) defined by\(\hat \chi\)Vg=χ(g)Vg\(\hat \chi\) (resp. the same for\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{\chi }\)). In sec. 4. the application of the foregoing results to the KMS-Algebra is given.